Generalized Yetter-Drinfel'd module categories for regular multiplier Hopf algebras

TL;DR

This paper extends the theory of Yetter-Drinfel'd modules for regular multiplier Hopf algebras by introducing generalized categories, exploring their structure as braided T-categories, and establishing isomorphisms in special cases.

Contribution

It introduces the generalized $(\alpha,eta)$-Yetter-Drinfel'd module categories and analyzes their properties as components of a braided T-category, expanding the framework for multiplier Hopf algebras.

Findings

Equivalence of Yetter-Drinfel'd module category to the centre of module category.

Introduction of generalized $(\alpha,eta)$-Yetter-Drinfel'd modules as braided T-category components.

Isomorphism between generalized categories and module categories over certain algebraic structures.

Abstract

For a regular multiplier Hopf algebra $A$, the Yetter-Drinfel'd module category ${}_{A}\mathcal{YD}^{A}$ is equivalent to the centre $Z({}_{A}\mathcal{M})$ of the unital left $A$-module category ${}_{A}\mathcal{M}$. Then we introduce the generalized $(\alpha, \beta)$-Yetter-Drinfel'd module categories ${}_{A}\mathcal{GYD}^{A}(\alpha, \beta)$, which are treated as components of a braided $T$-category. Especially when $A$ is a coFrobenius Hopf algebra, ${}_{A}\mathcal{YD}^{A}(\alpha, \beta)$ is isomorphic to the unital $\hat{A} \bowtie A(\alpha, \beta)$-module category ${}_{\hat{A} \bowtie A(\alpha, \beta)}\mathcal{M}$. Finally for a Yetter-Drinfel'd $A$-module algebra $H$, we introduce Yetter-Drinfel'd $(H, A)$-module category, which is a monoidal.